Anomalous quantum interference effects in graphene SNS junctions due to strain-induced gauge fields

We investigate the influence of gauge fields induced by strain on the supercurrent passing through the graphene-based Josephson junctions. We show that, in the presence of a constant pseudomagnetic field ${\mathbf{B}}_S$ originating from an arc-shape elastic deformation, the Josephson current is monotonically enhanced. This is in contrast with the oscillatory behavior of supercurrent (known as Fraunhofer pattern) caused by real magnetic fields passing through the junction. The absence of oscillatory supercurrent originates from the fact that strain-induced gauge fields have opposite directions at the two valleys due to the time-reversal symmetry. Subsequently the Aharonov-Bohm phases due to ${\mathbf{B}}_S$ cancel out between electron and hole components of the current carrying Andreev bound states. Nevertheless, we find another phase factor coming from the pseudomagnetic fields which rotates the pseudospin and is in fact responsible for the enhancement of Josephson currents. When both magnetic and pseudomagnetic fields are present, Fraunhofer-like oscillations as a function of the real magnetic field flux are found. Intriguingly, the combination of two kinds of gauge fields results in two special fingerprints in the local supercurrent density: (i) strong localization of the Josephson current density with more intense maximum amplitudes; (ii) appearance of the inflated vortex cores—finite regions with almost diminishing Josephson currents—whose sizes increase by increasing ${\mathbf{B}}_S$. These findings reveal unexpected interference signatures of pseudomagnetic fields in graphene SNS junctions which can cause a twist in the investigations on strain-induced gauge fields.

Figure 1

(a) Schematic of the SGS Josephson junction with strained graphene at the middle. (b) The classical trajectories of Dirac excitations inside the normal graphene. The gauge fields are weak enough so that they cannot deflect the trajectories and only phase effects corresponding to them can come into play. The geometric parametrization of the trajectories and the normal region are shown.

Figure 2

Current-phase relation for the SGS junction under a uniform strain-induced gauge field of various strengths given by ${\mathrm{\Phi }}_S/{\mathrm{\Phi }}_0$. Increasing the applied gauge field leads to larger range of phase differences in which the Josephson current is suppressed. Simultaneously, the maximum values of the Josephson current increases monotonically with the gauge field. The geometric aspect ratio of junction is $W/L=1$ and the temperature is considered to be $T=0.05T_c$.

Figure 3

Variation of critical current with the strength of strain-induced gauge field for some aspect ratios of the junction. Panels (a) and (b) correspond to the temperatures $T=0.05T_c$ and $T=0.05T_c$. In general, the critical current monotonically increases with the gauge field, however the amplitude of variations is larger for smaller geometric aspect ratios.

Figure 4

Formation of Josephson vortices at the presence of magnetic flux ${\mathrm{\Phi }}_B/{\mathrm{\Phi }}_0=4$ in (a) an SNS junction and (b)–(d) SGS junctions subjected to different values of strain-induced gauge fields. Each column corresponds to the same situation with upper and lower panels indicating the intensity plot and the vector field of the local current density, respectively. While the generic form of the vortex pattern remains almost the same in all cases, the local current profile in SGS junctions has a more localized pattern. The localization is intensified at the presence and by increase of the gauge field. The parameters considered in all subplots are $W/L=1,\varphi =\pi /4$, and $T/T_c=0.95$.

Figure 5

Evolution of local current density profile with pseudomagnetic field in the presence of a constant magnetic flux ${\mathrm{\Phi }}_B/{\mathrm{\Phi }}_0=0.5$. While the trend from (a) to (c) indicates that only local current becomes more localized and the maximum values increase, the behavior in the presence of large gauge field $\left({\mathrm{\Phi }}_S/{\mathrm{\Phi }}_0=5\right)$ is drastically different. In the last situation a big halo appears on the top of the junction where the supercurrent is almost diminished. The halo region has sharp boundaries with its surroundings in contrast to the conventional vortex cores (like the one on the lower edge) where the current density varies smoothly with the distance from the vortex center. The other parameters used here are the same as those in Fig. 4.

Figure 6

(a)–(d) Local current density profile inside the middle portion of a wide junction with $W/L=5$ at the presence of a constant total magnetic flux ${\mathrm{\Phi }}_B/{\mathrm{\Phi }}_0=2.5$ passing through the junction and different pseudomagnetic fluxes ${\mathrm{\Phi }}_S/{\mathrm{\Phi }}_0=0,5,10,15$, respectively. The portion is extended from $y/L=2$ to $y/L=3$ along vertical direction while it fills the whole length of the junction $\left(0<x/L<1\right)$. Compared to Fig. 5 the current profile is more localized along the central vertical line but it shows qualitatively similar behavior by increasing ${\mathrm{\Phi }}_S$ as before. It should be noted that the flux passing through the portion is $1/5$ of the total flux which makes the comparison with those in Fig. 5 reasonable. (f) Shows the local current throughout the junction when ${\mathrm{\Phi }}_S/{\mathrm{\Phi }}_0=0$ in correspondence to panel (a). It clearly shows an almost periodic variation of the current along vertical direction. All other parameters used here are the same as those in Fig. 5.

Figure 7

Magnetic oscillations of the critical Josephson current known as Fraunhofer pattern when the graphene junction is subjected to the pseudomagnetic fields of different strengths ${\mathrm{\Phi }}_S/{\mathrm{\Phi }}_0=0,1,3,5$. Panels (a) and (b) correspond to different width to length ratios of the junction $W/L=1,2$, respectively. While the maximum value of the critical current at ${\mathrm{\Phi }}_B=0$ increases monotonically with the pseudomagnetic field, its effect is less apparent at larger magnetic fluxes. By increasing the width of the junction, oscillations become more regular while aperiodic behavior appears when $W\lesssim L$.